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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppcnlem5 | Structured version Visualization version GIF version |
Description: Lemma for knoppcn 32619. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
Ref | Expression |
---|---|
knoppcnlem5.t | ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) |
knoppcnlem5.f | ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) |
knoppcnlem5.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
knoppcnlem5.1 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
Ref | Expression |
---|---|
knoppcnlem5 | ⊢ (𝜑 → (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))):ℕ0⟶(ℂ ↑𝑚 ℝ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | knoppcnlem5.t | . . . . . 6 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
2 | knoppcnlem5.f | . . . . . 6 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) | |
3 | knoppcnlem5.n | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
4 | 3 | ad2antrr 762 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑧 ∈ ℝ) → 𝑁 ∈ ℕ) |
5 | knoppcnlem5.1 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
6 | 5 | ad2antrr 762 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑧 ∈ ℝ) → 𝐶 ∈ ℝ) |
7 | simpr 476 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ) | |
8 | simplr 807 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑧 ∈ ℝ) → 𝑚 ∈ ℕ0) | |
9 | 1, 2, 4, 6, 7, 8 | knoppcnlem3 32610 | . . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧)‘𝑚) ∈ ℝ) |
10 | 9 | recnd 10106 | . . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧)‘𝑚) ∈ ℂ) |
11 | eqid 2651 | . . . 4 ⊢ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)) = (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)) | |
12 | 10, 11 | fmptd 6425 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)):ℝ⟶ℂ) |
13 | cnex 10055 | . . . . 5 ⊢ ℂ ∈ V | |
14 | reex 10065 | . . . . 5 ⊢ ℝ ∈ V | |
15 | 13, 14 | pm3.2i 470 | . . . 4 ⊢ (ℂ ∈ V ∧ ℝ ∈ V) |
16 | elmapg 7912 | . . . 4 ⊢ ((ℂ ∈ V ∧ ℝ ∈ V) → ((𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)) ∈ (ℂ ↑𝑚 ℝ) ↔ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)):ℝ⟶ℂ)) | |
17 | 15, 16 | ax-mp 5 | . . 3 ⊢ ((𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)) ∈ (ℂ ↑𝑚 ℝ) ↔ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)):ℝ⟶ℂ) |
18 | 12, 17 | sylibr 224 | . 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)) ∈ (ℂ ↑𝑚 ℝ)) |
19 | eqid 2651 | . 2 ⊢ (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))) = (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))) | |
20 | 18, 19 | fmptd 6425 | 1 ⊢ (𝜑 → (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))):ℕ0⟶(ℂ ↑𝑚 ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ↦ cmpt 4762 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 ↑𝑚 cmap 7899 ℂcc 9972 ℝcr 9973 1c1 9975 + caddc 9977 · cmul 9979 − cmin 10304 / cdiv 10722 ℕcn 11058 2c2 11108 ℕ0cn0 11330 ⌊cfl 12631 ↑cexp 12900 abscabs 14018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-sup 8389 df-inf 8390 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-fl 12633 df-seq 12842 df-exp 12901 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 |
This theorem is referenced by: knoppcnlem6 32613 |
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